Demonstration 1 of 4
The agenda picks the winner
With the same five ballots, can the order of votes decide the committee's choice?
When majority preference cycles, every plan loses to some other plan. The last plan introduced faces only the survivor, so the agenda setter chooses the outcome.
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Members 1 and 2 rank A > T > O; members 3 and 4 rank T > O > A; member 5's ranking is the control. In each round the survivor meets the next plan by simple majority.
Predict first. If member 5 switches to A > O > T, does the agenda order still matter?
Choose an example
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Constructed example: the chapter's hypothetical five-member committee and both rankings for member 5; all three agendas are from the book.
Calculated values
- A vs T
- 3 to 2
- T vs O
- 4 to 1
- O vs A
- 3 to 2
- First vote
- A vs T: A wins
- Winner
- outsourcing (O)
- Condorcet winner
- none (cycle)
Member 5 ranks O > A > T. First A meets T: 3 + 2 = 5 votes, A gets 3 and T gets 2, so A survives. Then A meets O, 2 to 3, so O wins. Majority preference cycles, so whoever sets the agenda picks the winner.
Worked steps
- A vs T: 3 to 2, A survives
- A vs O: 2 to 3, O wins
- Majority preference cycles, so whoever sets the agenda picks the winner.
Use the idea
Before trusting a sequence of majority votes, check every pairwise contest for a cycle.
Where the conclusion applies
Sincere voting, strict rankings and a fixed agenda. Strategic voters may vote against their ranking in early rounds.
Check your understanding: With member 5 at A > O > T and the agenda T vs O first, who wins?
Chapter 8 source: section "Condorcet paradox".
Demonstration 2 of 4
Single-peaked preferences restore a majority order
When does majority rule give a coherent ranking of three transport plans?
In the cyclic profile each plan beats one rival and loses to the other. On a common spending line with single peaks, the median voter's favourite beats every alternative and the majority relation is transitive.
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R is rail, B bus and D road repair, ordered by spending D < B < R. A profile is single-peaked when each voter's ranking falls away on both sides of a favourite on that line.
Predict first. In the single-peaked profile, if voter 3's favourite moves to B (B > R > D), does the majority winner change?
Choose an example
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Constructed example: the chapter's hypothetical cyclic and single-peaked transport profiles; voter 3 rankings B > R > D and D > B > R are added for comparison.
Calculated values
- Pairwise results
- R over B 2 to 1; B over D 2 to 1; D over R 2 to 1
- Single-peaked on D < B < R
- no
- Majority order
- cycle
- Majority winner
- none
Each contest splits the 3 votes, for example R vs B gets 2 + 1 = 3: R over B 2 to 1; B over D 2 to 1; D over R 2 to 1. So the majority relation cycles, so there is no majority order. Voter 3's control applies to the single-peaked profile only.
Worked steps
- R vs B: 2 + 1 = 3 votes, 2 to 1
- B vs D: 2 + 1 = 3 votes, 2 to 1
- D vs R: 2 + 1 = 3 votes, 2 to 1
- Order: cycle
Use the idea
Check whether the options lie on one dimension that every voter judges the same way before relying on majority rule.
Where the conclusion applies
Three sincere voters with strict rankings. Restricting the domain to single-peaked profiles gives up Arrow's unrestricted domain condition.
Check your understanding: In the single-peaked profile with voter 3 ranking D > B > R, what is the majority order?
Chapter 8 source: section "Arrow's impossibility theorem".
Demonstration 3 of 4
The median budget beats every challenger
Why does the median voter's ideal budget win every pairwise vote?
Whichever side the challenger is on, the median voter and every voter on the other side prefer 50. That is at least four of seven, a majority.
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Seven voters' ideal library budgets in millions of dollars. Each prefers the proposal closer to their ideal, u(x) = -|x - m|. The cutline is halfway between the two proposals.
Predict first. Does a far challenger like 90 lose by more than a near one like 70?
Choose an example
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Constructed example: the chapter's hypothetical seven voters (challengers 35 and 70); challengers 20 and 90 are added for comparison.
Calculated values
- Votes for 50
- 4
- Votes for the challenger
- 3
- Cutline
- 42.5
- Winner
- 50
Voters closer to 50 than to 35 are those on 50's side of the midpoint (50 + 35) / 2 = 42.5: 50, 65, 80 and 95. So 50 wins 4 to 3. The median's coalition always includes the median voter and everyone beyond, so 50 cannot lose.
Worked steps
- Cutline = (50 + 35) / 2 = 42.5
- Prefer 50: 50, 65, 80 and 95 = 4 voters
- Prefer 35: 3 voters
- Result 4 to 3
Use the idea
With one policy dimension and single-peaked preferences, expect proposals to converge on the median voter's ideal.
Where the conclusion applies
One dimension, sincere voting, single-peaked preferences and an odd number of voters.
Check your understanding: Against a $90 million challenger, what is the tally?
Chapter 8 source: section "Median voter theorem".
Demonstration 4 of 4
Rights plus unanimity can cycle
Can personal rights and the Pareto rule both be respected?
Each right forces one comparison and unanimity forces a third. When L cares about what P reads, the three close into a cycle; when L is indifferent about P's reading, the Pareto link goes away.
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In state a no one reads the book, in b person P reads it and in c person L reads it. P has the right to decide a versus b, L to decide a versus c. Pareto: if both prefer x to y, so does society.
Predict first. If the Pareto rule is switched off, does a transitive social order exist?
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Constructed example: the chapter's hypothetical disputed book (both views of L); switching the Pareto rule off is added for comparison.
Calculated values
- Required comparisons
- 3
- Cycle
- yes
- Consistent social order
- none
P's right over a versus b gives a > b; L's right over a versus c gives c > a. Both P and L rank b above c, so Pareto adds b > c. That makes 1 + 1 + 1 = 3 required comparisons, and c > a > b > c is a cycle, so no transitive social order exists.
Worked steps
- P's right: a > b
- L's right: c > a
- Both P and L rank b above c, so Pareto adds b > c.
- 3 comparisons: cycle
Use the idea
When rights and unanimous preferences conflict, look at which preferences are about other people's private choices.
Where the conclusion applies
Two people, three states and strict social comparisons from each rule.
Check your understanding: With L indifferent between a and b, which social order is consistent?
Chapter 8 source: section "Sen's liberal paradox".